c 2007 Cambridge University Press
J. Fluid Mech. (2007), vol. 593, pp. 385–404. doi:10.1017/S0022112007008889 Printed in the United Kingdom
385
The rise of Newtonian drops in a nematic
liquid crystal
C H U N F E N G Z H O U1 , P E N G T A O Y U E1,2
A N D J A M E S J. F E N G1,2
1
Department of Chemical and Biological Engineering, University of British Columbia, Vancouver,
BC V6T 1Z3, Canada
2
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
(Received 11 February 2007 and in revised form 21 August 2007)
We simulate the rise of Newtonian drops in a nematic liquid crystal parallel to
the far-field molecular orientation. The moving interface is computed in a diffuseinterface framework, and the anisotropic rheology of the liquid crystal is represented
by the Leslie–Ericksen theory, regularized to permit topological defects. Results reveal
interesting coupling between the flow field and the orientational field surrounding the
drop, especially the defect configuration. The flow generally sweeps the point and
ring defects downstream, and may transform a ring defect into a point defect. The
stability of these defects and their transformation are depicted in a phase diagram
in terms of the Ericksen number and the ratio between surface anchoring and bulk
elastic energies. The nematic orientation affects the flow field in return. Drops with
planar anchoring on the surface rise faster than those with homeotropic anchoring,
and the former features a vortex ring in the wake. These are attributed to the viscous
anisotropy of the nematic. With homeotropic anchoring, the drop rising velocity
experiences an overshoot, owing to the transformation of the initial surface ring
defect to a satellite point defect. With both types of anchoring, the drag coefficient
of the drop decreases with increasing Ericksen number as the flow-alignment of the
nematic orientation reduces the effective viscosity of the liquid crystal.
1. Introduction
Nematic liquid crystals exhibit special electro-optical properties and find applications in numerous modern technologies. As complex fluids, they are distinguished
microscopically by molecular alignment and long-range orientation order, and
macroscopically by a liquid–solid duality in that they flow as anisotropic viscous
fluids but resist orientational distortion as elastic solids (de Gennes & Prost 1993).
In a fluid-mechanical context, the motion of a particle or drop in a nematic is of
fundamental interest, being the counterpart of the Stokes or Hadamard–Rybczynski
problem in viscous Newtonian fluids. Besides, suspensions and emulsions in nematic
matrices show intriguing mesoscopic structures and mechanical properties that
suggest new applications (Poulin, Stark, Lubensky & Weitz 1997b; Poulin & Weitz
1998; Loudet, Barois & Poulin 2000; Tixier, Heppenstall-Butler & Terentjev 2006).
Particle motion in nematic liquid crystals is much more complex than the Stokes
problem. Even in a static nematic, insertion of a particle or drop normally causes the
nucleation of orientational defects (Poulin et al. 1997b; Lavrentovich 1998; Feng &
Zhou 2004). The liquid crystal molecules prefer a certain orientation on interfaces,
386
C. Zhou, P. Yue and J. J. Feng
the most common being homeotropic (normal) and planar (tangential) anchoring.
If the orientation field surrounding the drop or particle comes into conflict with
the far-field orientation, defects form. These may be seen as singularities in the
director field n(r), which represents the average molecular orientation at each spatial
point. For a particle with homeotropic anchoring, experiments have recorded two
types of defects: a ‘satellite’ point defect (Poulin et al. 1997b; Poulin & Weitz 1998;
Lubensky et al. 1998) and a ‘Saturn-ring’ line defect that encircles the particle on its
equator (Mondain-Monval et al. 1999; Gu & Abbott 2000). With planar anchoring,
two surface defects known as ‘boojums’ form at the poles (Poulin & Weitz 1998).
Orientational defects have long been an important subject of liquid crystal physics
and indeed condensed matter physics in general (Trebin 1982; Kléman 1983).
For moving particles, the earliest studies were falling-ball experiments to measure
the effective viscosity of liquid crystals (White, Cladis & Torza 1977; Kuss 1978). More
recently, Poulin et al. (1997a) used the ‘Stokes drag’ to verify the dipolar attraction
force between two droplets in a nematic fluid. In such dynamic situations, the flow
modifies the director field and defect configuration near the particle. The latter in turn
affect the rheology of the liquid crystal and thus the flow field. Therefore, the two-way
coupling between flow and microstructure is the key physics governing particle motion
in nematics. In general, such coupling has been formulated by constitutive theories for
nematic liquid crystals (de Gennes & Prost 1993; Rey & Tsuji 1998; Feng, Sgalari &
Leal 2000). Owing to the rheological complexity, only a handful of theoretical studies
have appeared on the moving particle problem, most of which sought to decouple
the flow field and the director field (Stark 2001). For instance, the director field may
be fixed at the static solution, and the resulting flow field and drag are calculated
(Ruhwandl & Terentjev 1996; Stark & Ventzki 2001). This corresponds to the low
Ericksen number (Er) limit, where the viscous forces are too weak to modify the
orientational field maintained by elasticity. Conversely, the Newtonian flow field may
be prescribed, and the director field n(r) is calculated as a result (Diogo 1983; Yoneya
et al. 2005). This may be linked to the high-Er limit. Stark & Ventzki (2002) seem
to have been the first to tackle the flow-director two-way coupling at finite Er. In
flow around a sphere with a satellite point defect, they predicted a counter-intuitive
flow effect that moves the defect upstream. This was contradicted by Yoneya et al.
(2005) who showed that the defect shifts downstream at a similar Ericksen number.
But the latter study prescribed the Stokes flow field, and it is unclear whether the
decoupling accounts for the discrepancy. To our knowledge, the only other coupled
study is Fukuda et al. (2004), who showed that the flow tends to convect the n field
downstream, while n modifies the velocity field and makes it fore–aft asymmetric.
Unfortunately, Fukuda et al. assumed an isotropic viscosity and thus omitted an
important component of the liquid crystal rheology. Therefore, a ‘complete solution’
that fully couples flow and director fields and incorporates viscous-elastic duality and
anisotropy is not yet available.
Most prior calculations assumed rigid anchoring on the particle surface. In reality,
the anchoring strength is finite, representable by an anchoring energy, and has a major
role in defining the defect configuration (Lubensky et al. 1998; Mondain-Monval
et al. 1999). Furthermore, only solid particles have been considered in theoretical
studies so far, even though most of the experimental observations have come from
emulsions with isotropic droplets suspended in nematics (Poulin & Weitz 1998). If the
interfacial tension is not so strong as to overwhelm the surface anchoring, the interplay
between the two is known to lead to unique drop and bubble shapes (Nastishin et al.
2005; Akers & Belmonte 2006; Zhou et al. 2007). These lacunae in our current
The rise of Newtonian drops in a nematic liquid crystal
387
understanding have motivated the present simulations based on the Leslie–Ericksen
theory.
Simulating the rise of Newtonian drops in a nematic liquid crystal is a computational
challenge because of the well-known numerical difficulties in handling moving and
deforming interfaces as well as the complex rheology of the nematic liquid crystal.
Not only are the rheological properties and stresses discontinuous across the interface,
they are anisotropic and evolving with the microstructure in the nematic component.
In principle, the balance between the stresses and the surface tension determines
the motion of the interface, which must be tracked dynamically on a moving grid
while solving for the flow in each component. As an alternative, we have developed
an energy-based diffuse-interface method that handles both the interface and the
rheology in a unified framework (Yue et al. 2004; Feng et al. 2005). The interface is
now a thin diffuse layer defined by a phase-field variable. A mixing energy governs
the interaction of the two components. As long as the microstructure of the complex
fluid is describable by a free energy, as is the case for liquid crystals, that energy
can be combined with the mixing energy to form the total free energy of the system.
A formal variational procedure then leads to the proper governing equations of the
two-fluid system. To solve these, Yue et al. (2006b) have developed two-dimensional
and axisymmetric finite-element algorithms based on adaptive mesh generation, which
is key to resolving the thin interface. The method has proved accurate and efficient
in simulating dynamics of viscoelastic drops and jets (Yue et al. 2005d; Yue, Zhou &
Feng 2006a; Zhou et al. 2006), and will be adapted to the problem at hand.
This study has three objectives: (1) to demonstrate that the motion of Newtonian
drops in a nematic fluid can be successfully simulated by our diffuse-interface method,
incorporating complex rheology, deformable interfaces and a variable anchoring
strength; (2) to investigate how flow modifies the orientational field and especially
the defect configuration near the drop; and (3) to investigate how the director field
modifies the flow in return, especially how the rheological anisotropy affects the rising
velocity of the drops and the drag force on them.
2. Theory and numerical method
Yue et al. (2004, 2006b) have described the theoretical model and the numerical
method in detail, and validated the methodology by benchmark problems. Planar
two-dimensional and axisymmetric applications to drop dynamics have been reported
recently (Yue et al. 2005a,b,c,d, 2006a; Zhou, Yue & Feng 2006). Therefore, we will
only summarize the main ideas and give the governing equations for a two-component
mixture of a Newtonian fluid and a nematic liquid crystal. The diffuse interface has a
small but non-zero thickness, inside which the two components are mixed and store
a mixing energy. We define a phase-field variable φ such that the concentrations of
the nematic and Newtonian components are (1 + φ)/2 and (1 − φ)/2, respectively.
Then φ = 1 in the bulk nematic phase, and φ = −1 in the bulk Newtonian phase.
The interface is taken to be the level set φ = 0. There are three types of free energies
in this system: mixing energy of the interface, bulk distortion energy of the nematic,
and the anchoring energy of the liquid crystal molecules on the interface:
λ
(φ 2 − 1)2
2
,
(2.1)
|∇φ| +
fmix =
2
2 2
K
(|n|2 − 1)2
,
(2.2)
∇n : (∇n)T +
fbulk =
2
2δ 2
388
fanch
C. Zhou, P. Yue and J. J. Feng
⎧ A
⎪
(planar anchoring)
⎨ (n · ∇φ)2
2
=
⎪
⎩ A [|n|2 |∇φ|2 − (n · ∇φ)2 ] (homeotropic anchoring).
2
(2.3)
√
In fmix , λ is the mixing energy density, is the capillary width and the ratio 2 2λ/3
produces the interfacial tension σ (Jacqmin 1999; Liu & Shen 2003; Yue et al. 2004);
fbulk is the Frank energy with a single elastic constant K. Different elastic constants
may be assigned to different modes of distortion (de Gennes & Prost 1993), but we
use the one-constant approximation for simplicity. Note that fbulk is regularized to
permit defects where |n| deviates from unity over the defect core of size δ. The second
term, devised by Liu & Walkington (2000) after the Ginzburg–Landau energy of
(2.1), represents the distortion energy of the defect by an energy penalty against the
shortening of |n|, effectively using |n| as an order parameter. For fanch , we adapt the
Rapini–Papoular form (Rapini & Papoular 1969) to our diffuse-interface
framework,
√
with A being the anchoring energy density and W = 2 2A/3 giving the surface
anchoring strength (Yamamoto 2001). Now we have the total free energy density for
the two-phase material:
1+φ
fbulk + fanch .
(2.4)
2
A variation on the free energy, supplemented by the various dissipative terms, leads
to the following governing equations for the configuration variables v, p, φ and n
(Yue et al. 2004):
f (φ, n, ∇φ, ∇n) = fmix +
ρ
∇ · v = 0,
∂v
+ v · ∇v = −∇p + ∇ · σ − ρgez ,
∂t
φ(φ 2 − 1)
∂φ
2
2
,
+ v · ∇φ = γ λ∇ −∇ φ +
∂t
2
h = γ1 N + γ2 D · n.
(2.5)
(2.6)
(2.7)
(2.8)
The density ρ is the average between the nematic density ρ1 and the Newtonian
density ρ2 :
1+φ
1−φ
ρ1 +
ρ2 ,
2
2
g is the gravitational acceleration and ez is the upward unit vector. The phase-field
variable φ obeys the Cahn–Hilliard equation, γ being the mobility parameter of the
diffuse interface (Yue et al. 2004; Yue, Zhou & Feng 2007). The deviatoric stress
tensor
1+φ 1−φ
1+φ
(∇n) · (∇n)T − G +
σ +
µ[∇v+(∇v)T ], (2.9)
σ = −λ(∇φ ⊗∇φ)−K
2
2
2
with G = A(n · ∇φ)n⊗∇φ for planar anchoring and G = A[(n · n)∇φ−(n · ∇φ)n]⊗∇φ
for homeotropic anchoring, and µ being the viscosity of the Newtonian component;
σ is the Leslie viscous stress in the nematic phase (Leslie 1968),
ρ=
σ = α1 D : nnnn + α2 nN + α3 Nn + α4 D + α5 nn · D + α6 D · nn,
(2.10)
where α1 to α6 are the Leslie viscous coefficients observing the Onsager relationship
(de Gennes & Prost 1993): α2 + α3 = α6 − α5 . D = (1/2)[∇v + (∇v)T ] is the strain
The rise of Newtonian drops in a nematic liquid crystal
389
rate tensor, and N = (dn/dt) − (1/2)[(∇v)T − ∇v] · n is the rotation of the director n
with respect to the background flow field. The n field evolves in the flow according
to a balance between elastic and viscous torques as given in (2.8). The elastic torque
is represented by the molecular field (de Gennes & Prost 1993):
1 + φ (n2 − 1)n
1+φ
∇n −
− g,
h=K ∇·
2
2
δ2
(2.11)
with g = A(n · ∇φ)∇φ for planar anchoring, and g = A[((∇φ · ∇φ)n − (n · ∇φ)∇φ]
for homeotropic anchoring. Both g and G derive from the anchoring energy fanch
through a variational procedure (Yue et al. 2004). The coefficients γ1 = α3 − α2 and
γ2 = α3 + α2 .
Obviously, the apparent viscosity of the nematic depends on the orientation of n
relative to the flow. This viscous anisotropy is commonly represented by the Miesowicz
viscosities in a simple shear flow (de Gennes & Prost 1993):
η1 = 12 (−α2 + α4 + α5 ),
η2 =
1
(α3
2
+ α4 + α6 ),
(2.12)
(2.13)
which are measured with n held perpendicular and parallel to the flow direction,
respectively; η1 > η2 . If n makes an angle θ with the flow, the general formula for
the shear viscosity is (Carlsson 1984):
η(θ) = η2 − (α2 + α3 ) sin2 θ.
(2.14)
A third Miesowicz viscosity may be defined with n along the vorticity axis. This
is irrelevant to the present study which constrains n to the meridional plane in
axisymmetric geometries. Without external fields or wall anchoring, n achieves a
steady alignment in simple shear if α2 /α3 > 0, but tumbles endlessly if the ratio is
negative. Most liquid crystals are of the aligning type (de Gennes & Prost 1993), and
the distinction is insignificant in complex flow fields as simulated here. Thus, we have
used α values based on the aligning PAA and MBBA in the rest of this paper.
To arrive at the Cahn–Hilliard equation (2.7), we have omitted from the right-hand
side coupling terms between fmix and the nematic energies. These are insignificant as
long as the interface stays narrow. In fact, the Cahn–Hilliard diffusive dynamics has
a visible effect only during singular events such as film rupture (Yue et al. 2005a). In
the current context, the diffuse interface may be seen as merely a numerical device
for treating a moving internal boundary.
The governing equations are solved, in axisymmetric geometries, by a numerical
scheme AMPHI that employs Galerkin finite elements with adaptive meshing (Yue
et al. 2006b). The latter has proved to be key to accurate phase-field simulations, and
the interface of thickness O(4) requires roughly 10 grid points to resolve (Feng et al.
2005). In addition, the defect regions are covered with fine grids as well (see figure 3b).
Since the size of the defect core is comparable to the interfacial thickness (Stark 1999),
δ is chosen to be 4. We use implicit time-stepping, with the time step automatically
adjusted according to the motion of the interface. Numerical experiments with grid
refinement and time-step refinement have been carried out (Yue et al. 2006b), and
adequate resolution is ensured for the simulations presented in the following.
390
C. Zhou, P. Yue and J. J. Feng
y
V
θ
L
–V
Figure 1. Simple shear flow of a nematic with homeotropic anchoring on the walls. The
director orientation is indicated by θ (y), and the velocity v(y) deviates from a linear profile
because of the anisotropic viscosity.
3. Results and discussion
3.1. Simple shear flow as validation
Consider the simple shear flow of a nematic between parallel walls in figure 1, with
rigid homeotropic anchoring on the walls. A one-dimensional analytical solution is
available if α1 vanishes, and we will compute the same flow in a two-dimensional
domain and use the exact solution to validate our numerical treatment of the Leslie–
Ericksen theory. In the one-dimensional solution due to Carlsson (1984), the velocity
profile v(y) and orientation profile θ(y) are given by coupled equations:
y
τ
dy − V ,
(3.1)
v(y) =
2
0 η2 − (α2 + α3 ) cos θ(y)
√
θ
K
1
√
dθ,
(3.2)
y(θ) = √
F
(θ)
−
F (θm )
2τ 0
2
√
θm
2
K
√
τ= 2
,
(3.3)
L
F (θ) − F (θm )
0
α2
η1
η2
−1
tan
F (θ) = √
+
tan θ − θ,
(3.4)
η1 η2
η2
η1
where τ is the constant shear stress on the plates determined from θm = θ(L/2), and
θm , the largest rotation angle at the centre between the walls, is in turn determined
by the condition V = v(L). This solution assumes α1 = 0, and is written in a slightly
simplified form here because of the one-constant approximation.
Our two-dimensional computation uses a domain of length 5L divided into
8235 triangular elements. Figure 2 compares our solution with Carlsson’s analytical
solution, and the two are in excellent agreement. Note that the maximum director angle
θm = 77.5◦ is short of the Leslie angle θ0 = 80◦ because at Er = η̄V L/K = 35, the
viscous effect is not strong enough to completely dominate the elastic effect. In Er, the
characteristic viscosity of the nematic is taken to be η̄ = (η1 + η2 )/2 = (α3 + α4 + α5 )/2.
The rotation of n into the flow direction reduces the local viscosity, and the velocity
profile v(y) reacts by diminishing the shear rate at the walls and increasing it in the
391
The rise of Newtonian drops in a nematic liquid crystal
θ
(a)
(b)
80
1.0
60
0.5
Simulation
Carlsson
40
V
0
–0.5
20
–1.0
0
Simulation
Carlsson
0.2
0.4
0.6
y
0.8
1.0
0
0.2
0.4
0.6
0.8
1.0
y
Figure 2. Comparison between our results and the one-dimensional exact solution of Carlsson
(1984). (a) The director orientation profile; (b) the velocity profile. The Leslie coefficients are
/α4 = −1.78, α3 /α4 = −0.056, α5 /α4 = 1, α6 /α4 = −0.83, and the Leslie angle
α1 = 0, α2√
θ0 = tan−1 α2 /α3 = 80◦ . The flow velocity corresponds to an Ericksen number Er = 35.
centre to maintain a constant shear stress. In this case, the minimum viscosity at the
centre ηm is such that η1 /ηm = 13.4 and ηm /η2 = 2.55.
3.2. Static orientational defects
When a drop has planar anchoring on its surface, boojums are the only possible
defects, even when the drop is moving in the liquid crystal. With homeotropic
anchoring, on the other hand, multiple defect patterns may appear and interesting
transformations take place. Thus we will only consider homeotropic anchoring in
this subsection, examining defects surrounding stationary particles as a preface to the
flow-induced transformation discussed in the next subsection. A more or less coherent
picture has emerged about defects near a stationary particle (Ruhwandl & Terentjev
1997; Stark 2001; Feng & Zhou 2004). The Saturn ring and the satellite point defect
are the two possible configurations (figures 3a and 3c), and their stability depends on
the relative importance of surface anchoring and bulk elasticity, represented by the
dimensionless group
Wa
,
(3.5)
AK =
K
a being the effective radius of the drop. A Saturn ring incurs more distortion to the
surface anchoring while a satellite costs more bulk energy. Thus, rings are favoured
at smaller AK . Indeed, the point defect becomes unstable below a critical AK , and
spontaneously opens into a Saturn ring. For sufficiently weak anchoring, the Saturn
ring shrinks onto the particle surface or even into the particle as an ‘imaginary
ring’ (Kuksenok et al. 1996). For larger AK , both point and ring defects are stable,
and either can be realized from proper initial conditions. The point defect becomes
energetically more favourable with increasing AK , but it is unclear whether the ring
ever becomes unstable (Ruhwandl & Terentjev 1997; Feng & Zhou 2004).
In regularizing the Leslie–Ericksen theory to allow defects (2.2), we treat n(r)
as a vector field. In reality, the molecular orientation is a pseudo-vector that does
392
C. Zhou, P. Yue and J. J. Feng
(a)
(b)
(c)
(d)
Figure 3. Defect configurations near a drop with homeotropic anchoring. (a) The satellite
point defect, indicated by the black dot, within the n field. Aσ = 0.05, AK = 100. (b) The
finite-element mesh for (a) is refined around the interface and the satellite defect. (c) Drawing
of the director field for a Saturn-ring defect. (d ) The surface ring defect for Aσ = 0.05,
AK = 100, indicated by black dots on the equator of the drop.
not distinguish n and −n. As a consequence, our vector-based theory cannot allow
defect lines of half-strength; the surrounding n field inevitably contains an apparent
discontinuity between n and −n and thus incurs an infinite elastic energy. Instead
of the Saturn-ring defect with a strength of −1/2, therefore, we predict a surface
ring as shown in figure 3(d ). With this caveat, we reproduce all the features noted
above, including the effect of AK on defect stability. In particular, for sufficiently
strong anchoring, the satellite defect is produced if the initial n is radial near the drop
surface, while the surface ring arises from an initially uniform n field. An additional
parameter,
W
(3.6)
Aσ = ,
σ
governs the shape of the drop, and a small value is used in most of the simulations to
ensure a nearly spherical drop. Figure 4 plots the position of the satellite point defect
as a function of AK . The increase of anchoring energy moves the point defect farther
from the drop, as is noted by Ruhwandl & Terentjev (1997). At the limit of AK → ∞,
rd /a → 1.35, which
√ agrees well with prior calculations (rd /a = 1.26 by Lubensky
et al. 1998 and 2 by Pettey et al. 1998) and measurement (rd /a = 1.4 ± 0.1 by
The rise of Newtonian drops in a nematic liquid crystal
393
1.40
1.35
1.30
rd
a
1.25
1.20
1.15
101
102
103
104
AK
Figure 4. Position of the satellite point defect near a stationary drop with homeotropic
anchoring. rd is the distance between the defect core and the centroid of the drop. For
AK < 10, the point defect loses stability and gives way to a surface ring on the equator.
Aσ = 0.05.
Cluzeau et al. 2001). With decreasing AK , the defect approaches the drop and causes
a protrusion on the interface. As AK falls below a threshold value, around 10 in
this case, the point defect opens up into a surface ring. This threshold AK is close
to the previous Monte Carlo prediction of approximately 7 (Ruhwandl & Terentjev
1997). The distance rd has some practical implications. Potentially it can be used as a
measurement of the anchoring strength W , which is otherwise difficult to determine.
Furthermore, rd also determines the particle spacing in self-assembled arrays of
droplets in nematic emulsions (Poulin & Weitz 1998).
3.3. Flow-induced transformation of defect configuration
Static defects may be driven from one configuration to the other by an external electric
or magnetic field (Terentjev 1995; Loudet & Poulin 2001). It will be interesting to
see whether similar transitions can be effected by the flow surrounding a rising drop.
Figure 5 shows schematically the computational domain for simulating rising drops
in a nematic medium. A spherical drop of radius a is initially centred at (0, 4a), with
either homeotropic or planar anchoring, although the latter will not be discussed
until the next subsection. The far-field director orientation is vertical and parallel to
the drop motion. A horizontal far-field n would upset axisymmetry and require a
fully three-dimensional simulation. We also disallow azimuthal components of n and
v. Thus, n = (0, 1) on the bottom, side and top walls. The velocity vanishes on the
bottom and sidewalls, but the top is assigned a stress-free condition. On the axis of
symmetry, we require ∂/∂r = 0 for all variables except the radial components of n and
v: nr = 0 and vr = 0. For numerical parameters, the capillary width = 0.01a, and a
small grid size h1 = 0.006a is used on the interface and near the defect (cf. figure 3b).
Inside the drop and in the matrix, the grid sizes are h2 = 0.08a and h3 = 0.1a,
respectively. These prescribed values are guidelines for mesh generation, and the
actual mesh is spatially unstructured and varies adaptively during the simulation.
394
C. Zhou, P. Yue and J. J. Feng
Z
n
H
a
r
L
Figure 5. Computational domain for a Newtonian drop rising in a nematic whose far-field
orientation is vertical. The geometry is axisymmetric and only half of the meridian plane is
used in the computation. The drawing is not to scale; H = 20a and L = 10a.
The rise of drops or bubbles in a nematic fluid is governed by 10 dimensionless
numbers:
ρ2
(drop-to-matrix density ratio),
(3.7)
α=
ρ1
µ
(drop-to-matrix viscosity ratio),
(3.8)
β=
η̄
ρga 2
Eo =
(Eótvös number),
(3.9)
σ
ρgη̄4
(Morton number),
(3.10)
Mo = 2 3
ρ1 σ
plus the 4 ratios of the 5 independent Leslie viscosities and the static parameters
Aσ and AK . In Eo and Mo, ρ = ρ1 − ρ2 . Drop and bubble shapes deviate from
the spherical at large Eo (Grace, Wairegi & Nguyen 1976), and our code has been
shown to accurately capture this effect for Newtonian fluids (Yue et al. 2006b). For
all results presented hereafter, α = 0.5 and β = 0.514. A set of Leslie coefficients
based on those of PAA (Chandrasekhar 1992) and MBBA (de Gennes & Prost
1993) are chosen as the baseline. Values of α1 , α3 , α4 and α5 are fixed at the ratio
of α1 : α3 : α4 : α5 = 5 : −1 : 18 : 18. This also fixes the characteristic viscosity
η̄ = (α3 + α4 + α5 )/2. Then the viscosity ratio η1 /η2 is varied through α2 (and α6
according to the Onsager relationship) to probe the effect of viscous anisotropy. Based
on the terminal velocity U of the rising drop, we determine the steady-state Reynolds
number Re = ρ1 U a/η̄ and Ericksen number Er = η̄U a/K.
Our results show that the flow shifts the orientation field downstream, and modifies
the relative stability of the ring and point defects. The ‘phase diagram’ in figure 6
depicts the stability of each configuration near a steadily rising drop, parametrized
by Er and AK that denote respectively the strengths of flow and surface anchoring
as compared with the bulk elasticity. Six zones may be identified with different defect
The rise of Newtonian drops in a nematic liquid crystal
102
VI
Point defect
Ring defect
Both defect
V
IV
101
Er
100
395
II
I
III
10–1
10–2
100
101
102
AK
Figure 6. A ‘phase diagram’ of steady-state defect configurations. AK and Er are varied by
tuning K and g, respectively. Aσ = 0.01, η1 /η2 = 3.
configurations. In zone I, the drop has an imaginary ring inside but no defects outside.
In zone II, the surface ring is the sole stable configuration, while in zone IV, only
point defects occur. In zone III, V and VI, both the surface ring and the satellite
point defect are locally stable and either may appear depending on initial conditions.
For vanishing Er, the transitions from zone I to II and III with increasing anchoring
strength are well known from static studies (Kuksenok et al. 1996; Ruhwandl &
Terentjev 1997). This simple picture holds up to Er ∼ 1. In this weak-flow regime,
the only flow effect is to shift the surface ring or satellite defect downstream. The
shift is more pronounced for higher AK , since stronger surface anchoring favours a
smaller ring. For higher Er, a transition from zone II to V or from zone III to IV
takes place depending on AK . In zone V, the point defect becomes locally stable; an
initial point defect can now be stabilized in the wake of the drop by a sufficiently
strong flow whereas in zone II, it would have opened up into a ring defect. If the
initial condition has a ring defect, it remains stable in zone V but shifts downstream
and shrinks in radius with increasing Er.
Going from zone III to IV with increasing Er, the ring defect loses stability. On
a drop that initially bears a surface ring, the flow sweeps the ring downstream on
the drop as it rises. If its terminal velocity puts it in zone IV, the surface ring will
be shed into a satellite point defect in the wake. Starting with an initial point defect,
we have only simulated the configuration with the defect in the wake of the drop.
Having a point defect upstream of the drop appears unlikely in reality and may
indeed be unstable to three-dimensional disturbances. Throughout zone III and zone
IV, the point defect remains stable and shifts downstream with increasing Er. The
steady-state position of the ring and point defects is shown in figure 7 as a function
of Er that crosses from zone III to IV.
Increasing Er further from zone IV, there is another transition to zone VI where the
surface ring regains stability. At such high Er, the drop assumes an oblate shape, and
an initial equatorial ring turns into a small surface ring near the bottom that cannot
be shed into the wake as a point defect. Figure 8 shows an example of the steady-state
director field in zone VI. Thanks to the oblate shape, the flow near the rear stagnation
point of the drop is much reduced as compared with zone IV. The viscous forces are
thus much weaker and can no longer drive the surface ring off. Similar configurations
396
C. Zhou, P. Yue and J. J. Feng
Zone III
Zone IV
1.5
Point defect
Ring defect
rd 1.0
a
0.5
0
10–1
100
101
Er
Figure 7. Steady-state position of the defect near a rising drop as a function of the Ericksen
number. rd is the distance between the centroid of the drop and the point defect (rd /a > 1) or
the centre of the surface ring (rd /a < 1). The arrow indicates the transition from zone III to
IV at Er = 1.10 when the surface ring defect gives way to a point defect. Aσ = 0.01, AK = 30,
η1 /η2 = 3.
Figure 8. Director orientation around a steadily rising oblate drop in zone VI, with a small
surface ring indicated by two black dots in the rear of the drop. η1 /η2 = 3, Aσ = 0.01,
AK = 20, Er = 53.6, Eo = 1.20, Mo = 1.22 × 10−6 and Re = 21.9.
occur at high Er in zone V. In fact, zone V and VI are connected at the top, and their
division is mostly a result of our describing the phase diagram in terms of increasing
Er from equilibrium. The multiplicity of defect configurations in zones III, V and VI
implies hysteresis. For example, raising Er transforms a ring in zone III into a point
defect in IV. Upon lowering Er back into zone III, however, the point defect remains
(cf. figure 7).
The trend in figure 7, showing defects being ‘convected’ downstream by flow, agrees
with the results of Yoneya et al. (2005) and Fukuda et al. (2004), but contradicts
the prediction of Stark & Ventzki (2002) that defects shifts upstream under flow.
As defects are orientation patterns rather than material properties, the convective
effect is not intuitively obvious. Thus, we designed an experiment using silicone oil
397
The rise of Newtonian drops in a nematic liquid crystal
(a)
(b)
0.5
0.30
0.4
0.25
0.3
V
0.20
η1/η2 = 34
η1/η2 = 17
η1/η2 = 4.7
η1/η2 = 2
0.2
0.1
η1/η2 = 34
η1/η2 = 17
η1/η2 = 4.7
η1/η2 = 2
0.15
0.10
0
10
20
30
t
40
50
0
20
40
60
t
Figure 9. Transient rising velocity of a drop in a nematic liquid crystal with (a) planar
anchoring and (b) homeotropic anchoring. Eo = 0.3, Mo = 5.56 × 10−5 , Aσ = 0.2 and
AK = 40. Time is made dimensionless by η̄/(ρga), and velocity by ρga 2 /η̄.
drops rising in the nematic 5CB (Khullar, Zhou & Feng 2007). Both the Saturn ring
and the point defect shift downstream as the rising velocity increases. At sufficiently
high speed, the Saturn ring is shed into the wake as a point defect. This settles the
question of the direction of convection, and confirms the flow-induced ring-to-point
transformation predicted here. The experimental conditions correspond to AK ≈ 25
and Er ≈ 0.25, comparable to the values in figure 7. Note that Yoneya et al. (2005)
have predicted a similar transformation at Er ∼ 10, but with the flow field prescribed
as the Stokes solution and with rigid anchoring (AK → ∞).
We should mention that the ranges of dimensionless parameters in the preceding
discussion correspond to common small-molecule nematics under reasonable flow
conditions, and the same is true for the next section. For example, the Leslie viscosity
ratios are close to those of flow-aligning nematics such as PAA and MBBA. Take
MBBA (de Gennes & Prost 1993): η̄ = 7.25×10−2 Pas, K = 5×10−12 N (average of the
three elastic constants for splay, twist and bend). Then the range of 0.1 . Er . 100
in figures 6 and 7 corresponds to rising velocities from 0.14 to 140 µm s−1 for a drop
of diameter 100 µm, which are consistent with the experimental values of Khullar
et al. (2007).
3.4. Rising velocity, drag force and the flow field
In this subsection, we investigate the effect of the nematic microstructure on the flow
field, with special attention to the implications of the viscous anisotropy and defect
configuration. The geometric setup of the computation is the same as in the last
subsection (cf. figure 5), but some of the parameter values differ. In particular, we
will focus on a range of rise velocity that corresponds to zone IV for homeotropic
anchoring, with the satellite defect being the sole stable configuration. This range
displays the most interesting behaviour when a ring defect transforms into a satellite
during the rise of the drop. Planar anchoring will be considered as well. Figure 9
shows the transient rising velocity of a drop with planar and homeotropic anchoring.
To give a sense of the time and velocity scales, a silicone oil drop 100 µm in diameter
rising in MBBA would have η̄/(ρga) = 1.54 s and ρga 2 /η̄ = 32.5 µm s−1 , both
398
C. Zhou, P. Yue and J. J. Feng
(a)
(b)
(c)
(d)
Figure 10. Director and flow fields around the drop with homeotropic anchoring at three
times: (a) t = 6.33, (b) t = 28.4, and (c) final steady state at t = 52.7, with Er = 15.8 and
Re = 6.43. (d ) The steady state for a drop with planar anchoring; Er = 25.2 and Re = 10.3.
These correspond to the curves in figure 9 with η1 /η2 = 34. For the streamlines, the reference
frame is affixed to the centroid of the drop.
within the experimental range of Khullar et al. (2007). Within each plot, we examine
the effects of viscous anisotropy by varying the ratio of two Miesowicz viscosities. As
explained before, this is achieved by varying α2 and α6 while keeping the characteristic
viscosity η̄ = (η1 + η2 )/2 and the other Leslie coefficients fixed.
With planar anchoring, the rising velocity V increases monotonically in time toward
the terminal velocity U . For homeotropic anchoring, on the other hand, V experiences
an overshoot. This is caused by the transition from a surface ring to a point defect
as explained in the last subsection. Figure 10 shows the director and flow fields near
the drop with homeotropic anchoring at three times. Initially, n is vertical throughout
the domain. A surface ring forms quickly on the equator of the drop and shifts
downstream as the drop rises (t = 6.33). With the drop accelerating, the flow sweeps
the defect ring towards the rear of the drop (t = 28.4), and eventually transforms
it into a point defect as V attains the terminal velocity (t = 52.7). Comparing
figures 10(b) and 10(c), the n field with the point defect has a larger area – including
the wake – in which n are nearly orthogonal to the streamlines. According to (2.14),
the nematic exhibits higher viscosity there than in areas where n is aligned to the
flow. Thus, the transformation from surface ring to point defect increases the viscous
dissipation in the entire domain and thus the drag on the drop. This explains the
overshoot in figure 9(b). Our recent experiment has confirmed such an overshoot
during the ring-to-point defect transformation (Khullar et al. 2007). If the drop has
a point defect at the start, or if the steady state falls in zones III, V or VI where
the surface ring is stable, there will be no overshoot in V . For planar anchoring, two
The rise of Newtonian drops in a nematic liquid crystal
399
0.5
0.4
U
0.3
0.2
Planar anchoring
Homeotropic anchoring
100
101
η1/η2
102
Figure 11. Terminal velocity U of the drops in figure 9 as affected by viscous anisotropy. U
is made dimensionless by ρga 2 /η̄. The range of U corresponds to 4.93 < Re < 10.9 and
12.1 < Er < 26.8 for planar anchoring, and 4.29 < Re < 6.64 and 10.5 < Er < 16.3 for
homeotropic anchoring.
boojums stay at the poles throughout the rise (figure 10d ), and the rising velocity
again shows no overshoot. Roughly speaking, the orientation distortion extends into
the nematic bulk for a fraction of the drop diameter; within this region the flow is
affected by the anisotropic viscosity.
With either type of anchoring, the rise velocity increases with viscous anisotropy as
measured by η1 /η2 . Figure 11 gives a clearer view of this effect in terms of the terminal
velocity U . Furthermore, planar anchoring produces a higher U than homeotropic
anchoring under otherwise identical conditions. This difference can again be explained
by the drag as affected by different director fields. With planar anchoring (figure 10d ),
n aligns with the streamlines in most of the domain, apparently minimizing the total
dissipation (Jadżyn & Czechowski 2001). This leads to a smaller drag and hence
a greater U than the drop with homeotropic anchoring. In both cases, however,
alignment between n and v is more prevalent throughout the domain than their
being orthogonal, and larger areas experience η2 than η1 . With increasing η1 /η2 ,
therefore, the overall viscous dissipation diminishes with η2 and the rising velocity
U increases as in figure 11, and more significantly for planar anchoring. In fact,
for each value of η1 /η2 , the U values for both anchoring types are bounded by the
Hadamard–Rybczynski predictions using η1 and η2 .
The eddies in the wake of the drop in figure 10(d ) form a vortex ring. It is not
expected for Newtonian fluids at Re = 10.3 and viscosity ratio β = 0.514 (Dandy &
Leal 1989), nor does it appear for homeotropic anchoring. The explanation seems to
rest with the anisotropic viscosity. The Reynolds number cited above is defined using
the characteristic viscosity η̄. In figure 10(d ), the streamlines align with the director
n to varying degrees in the flow field. Thus, the local viscosity may be much below η̄
in some regions. In particular, the streamlines next to the drop surface follow the n
field precisely. If we take the Miesowicz viscosity η2 to be the local viscosity, the local
Reynolds number will be around 180. The high momentum of this layer of fluid then
leads to flow separation in the wake. In contrast, homeotropic anchoring causes n to
be mostly orthogonal to v near the drop surface. The local Reynolds number is much
lower and no separation occurs. We have also noticed that for planar anchoring,
400
C. Zhou, P. Yue and J. J. Feng
Er
10–1
100
101
14
Homeotropic anchoring
Planar anchoring
CD∗Re
12
10
8
10–1
100
101
Re
Figure 12. The drag coefficient for drops rising in a nematic as a function of Re or Er. For
homeotropic anchoring, the data correspond to the satellite configuration. η1 /η2 = 3, Aσ = 0.2,
AK = 40.
the recirculating zone shrinks with decreasing η1 /η2 and disappears altogether when
η1 = η2 . This is consistent with viscous anisotropy being the cause of the vortex
ring.
The drag on the drop can be further analysed in terms of the drag coefficient
CD =
4
πρga 3
3
1
ρ U 2 (πa 2 )
2 1
(3.11)
defined from the terminal velocity U . The Hadamard–Rybczynski formula gives
CD Re = 8(1 + 1.5β)/(1 + β) for a spherical Newtonian drop moving in a Newtonian
matrix at vanishing Re (Batchelor 1980). In view of this formula, we plot the product
CD Re against Re and Er in figure 12. U and hence Re and Er are varied through the
buoyancy force while keeping the viscosity ratio β fixed. As noted before, homeotropic
anchoring gives a higher drag than planar anchoring. If the matrix were a Newtonian
fluid, CD Re would be constant for small Re and increase with Re for finite inertia. That
CD Re decreases with increasing Reynolds number reflects the enhanced alignment of
n by the flow field. This is better illustrated by Er marked on the upper abscissa.
For small Ericksen numbers, say Er < 1, the director orientation is hardly affected
by the surrounding flow. Thus CD Re remains roughly constant. As Er exceeds unity,
viscous flow effects become comparable to the elastic effects and the flow-alignment
of n reduces CD Re. This decline eventually levels off as the flow-alignment saturates
around Er = 10.
All prior results in the literature on the drag force are for rigid spheres with
homeotropic anchoring at Re = 0. Nevertheless, a comparison is interesting. In the
limit of vanishing Er, Stark & Ventzki (2001) fixed the n(r) field to the equilibrium
solution with a point defect, and computed the drag in terms of an effective viscosity
ηeff defined from the Stokes formula. In our case, a similar ηeff can be estimated
from the Hadamard–Rybczynski formula. At the lowest Er = 0.109 for homeotropic
anchoring, our data give ηeff /η2 = 2.63, which is comparable to the results of Stark
& Ventzki (2001): 1.83 for MBBA and 2.32 for 5CB. Our somewhat larger drag may
The rise of Newtonian drops in a nematic liquid crystal
(a)
(b)
401
(c)
Figure 13. Non-spherical drop shapes produced by the nearby defects. Eo = 0.3,
Mo = 5.56 × 10−5 , Aσ = 0.5, AK = 15, η1 /η2 = 4.7. (a) A drop with homeotropic anchoring
and a surface ring rising at Re = 5.54, Er = 2.04. (b) A drop with homeotropic anchoring
and a point defect. Re = 3.92, Er = 1.44. (c) A drop with planar anchoring and boojums.
Re = 7.16, Er = 2.64. The boojums give the highest rising velocity while the point defect the
lowest.
have to do with wall confinement in the geometric setup (figure 5). The decrease of
ηeff with Er is consistent with the findings of Stark & Ventzki (2002). At the highest
Er = 14.1 in figure 12, ηeff /η2 = 2.22. Although the Hadamard–Rybczynski formula
no longer applies exactly at the finite Re in this case, ηeff asymptoting to a value
significantly larger than η2 bespeaks the ‘orientation boundary layer’ on the drop
surface due to the homeotropic anchoring (cf. figure 10c), inside which there is still
considerable misalignment between n and v. With even higher Er, the experiment of
White et al. (1977) suggests that ηeff approaches η2 . For planar anchoring, our data
give ηeff /η2 = 2.18 and 1.60 for the low-Er and high-Er limits of figure 12.
So far we have used relatively large values for the interfacial tension σ (or small
values of Aσ ) to keep the surface curvature of the drop smooth. At larger Aσ ,
drops deform in response to the nearby orientational field, especially the presence of
defects. Figure 13 illustrates three typical situations with the surface ring, satellite and
boojum defects. The proximity of defects causes large curvature on the drop surface
as a result of the competition between interfacial tension and anchoring energy. The
cost in anchoring energy due to the defects is reduced at the expense of interfacial
area such that the total energy is minimized. The lemon shape in figure 13(c) is
well known in nematic drops with planar anchoring, and has also been reported for
isotropic drops in nematic medium (Nastishin et al. 2005; Zhou et al. 2007). The drop
with boojums rises the fastest while that with the satellite the slowest.
4. Summary
This paper presents a computational study of the rise of a Newtonian drop in
a nematic liquid crystal. The problem is a rough counterpart of the Hadamard–
Rybczynski problem in Newtonian fluids, although the Reynolds number ranges
up to about 10 and mild drop deformation occurs. The key physics revealed by
the simulation is the two-way coupling between the flow field and the molecular
orientation field, and especially the configuration of orientational defects. The results
can thus be summarized as follows.
(a) Effect of flow on the orientational field. With either a satellite point defect
or a surface ring defect, the flow sweeps the defect downstream. Thus, the surface
ring shrinks and moves toward the rear stagnation point, and at sufficiently high
402
C. Zhou, P. Yue and J. J. Feng
Ericksen number may be transformed into a point defect. An initial point defect
moves farther downstream with increasing Ericksen number. The stability of the two
defect configurations is depicted by a phase diagram in terms of the Ericksen number
and the ratio between surface anchoring and bulk elastic energies.
(b) Effect of orientation on the flow field. This is mainly manifested through the
viscous anisotropy of the fluid. Drops with planar anchoring rise faster than those with
homeotropic anchoring since the director field is better aligned with the streamlines.
With homeotropic anchoring, a drop experiences an overshoot in the transient rising
velocity when a ring defect changes into a detached point defect. With both types of
anchoring, the drag coefficient decreases with the Ericksen number because stronger
viscous flow aligns the director to the streamline and reduces frictional dissipation.
Through a systematic examination of the coupling between flow and molecular
orientation, we strive to construct a coherent picture for the fluid mechanics of
a particle moving in a nematic liquid. So far, the predicted flow effects on defect
convection and transformation have been verified experimentally, as has the overshoot
in rise velocity accompanying the defect transformation (Khullar et al. 2007). Finally,
we point out two limitations in our work. The first is the vectorial nature of the Leslie–
Ericksen theory. The original version cannot handle defects as they would constitute
singularities. A relaxation of the unit-length requirement on the director allows
integer-strength defects to be simulated, but the Saturn ring has to be represented by
a surface ring. The latter has fewer degrees of freedom and possibly different stability
regimes from an unattached Saturn ring. This restriction can be removed by adopting
a tensorial representation of the molecular orientation (Rey & Tsuji 1998; Feng et al.
2000; Yoneya et al. 2005). Second, the axisymmetric two-dimensional geometry of
the computation precludes the interesting scenario of drops rising in a nematic with
horizontal far-field orientation.
Acknowledgement is made to the Donors of The Petroleum Research Fund,
administered by the American Chemical Society, for partial support of this research.
J.J.F. was also supported by the NSERC, the Canada Research Chair program,
the Canada Foundation for Innovation and the NSFC (Grant no. 50390095). C.Z.
acknowledges partial support by a University Graduate Fellowship from UBC.
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